Chapter 3: Numerical Descriptive Measures
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers
the question.
1) Which of the following statistics is not a measure of central tendency?
A) Arithmetic mean
B) Q3
C) Mode
D) Median
1)
2) Which of the arithmetic mean, median, mode, and geometric mean are
resistant measures of central tendency?
A) The arithmetic mean and median only
B) The mode and arithmetic mean only
C) The median and mode only
D) The arithmetic mean and mode only
2)
3) Which of the following is the easiest to compute?
A) The sum of the values
B) The arithmetic mean
C) The mode
D) The median
3)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers
the question.
SCENARIO 3-1
Health care issues are receiving much attention in both academic and political arenas. A sociologist
recently conducted a survey of citizens over 60 years of age whose net worth is too high to qualify
for Medicaid. The ages of 25 senior citizens were as follows:
60 61 62 63 64 65 66 68 68 69 70 73 73
74 75 76 76 81 81 82 86 87 89 90 92
4) Referring to Scenario 3-1, calculate the arithmetic mean age of the
senior citizens to the nearest hundredth of a year.
4)
5) Referring to Scenario 3-1, determine the median age of the senior
citizens.
5)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers
the question.
6) True or False: The median of the values 3.4, 4.7, 1.9, 7.6, and 6.5 is 1.9.
A) True
B) False
1
6)
7) True or False: The median of the values 3.4, 4.7, 1.9, 7.6, and 6.5 is 4.05.
A) True
B) False
7)
8) True or False: In a sample of size 40, the sample mean is 15. In this case, the
sum of all observations in the sample is
X i = 600.
8)
9) True or False: The median of a data set with 20 items would be the average of
the 10th and the 11th items in the ordered array.
A) True
B) False
9)
A) True
B) False
SHORT ANSWER. Write the word or phrase that best completes each statement or answers
the question.
SCENARIO 3-3
The ordered array below represents the number of vitamin supplements sold by a health food
store in a sample of 16 days.
19, 19, 20, 20, 22, 23, 25, 26, 27, 30, 33, 34, 35, 36, 38, 41
Note: For this sample, the sum of the values is 448, and the sum of the squared di erences between
each value and the mean is 812.
10) Referring to Scenario 3-3, the arithmetic mean of the number of
vitamin supplements sold in this sample is ________.
10)
11) Referring to Scenario 3-3, the median number of vitamin supplements
sold in this sample is ________.
11)
SCENARIO 3-5
The rates of return of a Fortune 500 company over the past 15 years are: 3.17%, 4.43%, 5.93%,
5.43%, 7.29%, 8.21%, 6.23%, 5.23%, 4.34%, 6.68%, 7.14%, -5.56%, -5.23%, -5.73%, -10.34%
12) Referring to Scenario 3-5, compute the arithmetic mean rate of return
per year.
2
12)
SCENARIO 3-7
In a recent academic year, many public universities in the United States raised tuition and fees due
to a decrease in state subsidies. The change in the cost of tuition, a shared dormitory room, and the
most popular meal plan from the previous academic year for a sample of 10 public universities
were as follows: $1,589, $593, $1,223, $869, $423, $1,720, $708, $1425, $922 and $308.
13) Referring to Scenario 3-7, what is the mean and median change in the
cost?
13)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers
the question.
14) In a right-skewed distribution
A) the median is greater than the arithmetic mean.
B) the median equals the arithmetic mean.
C) the median is less than the arithmetic mean.
D) none of the above.
14)
15) True or False: The coefficient of variation measures variability in a data set
relative to the size of the arithmetic mean.
A) True
B) False
15)
16) True or False: The coefficient of variation is a measure of central tendency in
the data.
A) True
B) False
16)
17) True or False: The coefficient of variation is a measure of relative variation.
A) True
B) False
17)
3
SHORT ANSWER. Write the word or phrase that best completes each statement or answers
the question.
SCENARIO 3-1
Health care issues are receiving much attention in both academic and political arenas. A sociologist
recently conducted a survey of citizens over 60 years of age whose net worth is too high to qualify
for Medicaid. The ages of 25 senior citizens were as follows:
60 61 62 63 64 65 66 68 68 69 70 73 73
74 75 76 76 81 81 82 86 87 89 90 92
18) Referring to Scenario 3-1, calculate the skewness statistic for the age of
the senior citizens accurate to two decimal places.
18)
19) Referring to Scenario 3-1, what type of shape does the distribution of
the sample appear to have?
19)
20) Referring to Scenario 3-1, calculate the kurtosis statistic for the age of
the senior citizens accurate to two decimal places.
20)
21) Referring to Scenario 3-1, does the distribution of the sample appear
to be lepokurtic or platykurtic?
21)
22) Referring to Scenario 3-1, calculate the variance of the ages of the
senior citizens correct to the nearest hundredth of a year squared.
22)
23) Referring to Scenario 3-1, calculate the standard deviation of the ages
of the senior citizens correct to the nearest hundredth of a year.
23)
24) Referring to Scenario 3-1, calculate the coe cient of variation of the
ages of the senior citizens.
24)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers
the question.
25) True or False: A population with 200 elements has an arithmetic mean of 10.
From this information, it can be shown that the population standard
deviation is 15.
A) True
B) False
4
25)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers
the question.
SCENARIO 3-3
The ordered array below represents the number of vitamin supplements sold by a health food
store in a sample of 16 days.
19, 19, 20, 20, 22, 23, 25, 26, 27, 30, 33, 34, 35, 36, 38, 41
Note: For this sample, the sum of the values is 448, and the sum of the squared di erences between
each value and the mean is 812.
26) Referring to Scenario 3-3, the skewness statistic of the number of
vitamin supplements sold in this sample is ________.
26)
27) Referring to Scenario 3-3, is the number of vitamin supplements sold
in this sample right- or left- skewed?
27)
28) Referring to Scenario 3-3, the kurtosis statistic of the number of
vitamin supplements sold in this sample is ________.
28)
29) Referring to Scenario 3-3, is the number of vitamin supplements sold
in this sample lepokurtic or platykurtic?
29)
30) Referring to Scenario 3-3, the range of the number of vitamin
supplements sold in this sample is ________.
30)
31) Referring to Scenario 3-3, the variance of the number of vitamin
supplements sold in this sample is ________.
31)
32) Referring to Scenario 3-3, the standard deviation of the number of
vitamin supplements sold in this sample is ________.
32)
33) Referring to Scenario 3-3, the coe cient of variation of the number of
vitamin supplements sold in this sample is ________ percent.
33)
34) Referring to Scenario 3-3, what type of shape does the distribution of
the sample appear to have?
34)
5
SCENARIO 3-5
The rates of return of a Fortune 500 company over the past 15 years are: 3.17%, 4.43%, 5.93%,
5.43%, 7.29%, 8.21%, 6.23%, 5.23%, 4.34%, 6.68%, 7.14%, -5.56%, -5.23%, -5.73%, -10.34%
35) Referring to Scenario 3-5, what is the range of the rate of return?
35)
36) Referring to Scenario 3-5, what is the shape of the distribution for the
rate of return?
36)
SCENARIO 3-7
In a recent academic year, many public universities in the United States raised tuition and fees due
to a decrease in state subsidies. The change in the cost of tuition, a shared dormitory room, and the
most popular meal plan from the previous academic year for a sample of 10 public universities
were as follows: $1,589, $593, $1,223, $869, $423, $1,720, $708, $1425, $922 and $308.
37) Referring to Scenario 3-7, what is the standard deviation of the change
in the cost?
37)
38) Referring to Scenario 3-7, what is the coe cient of variation of the
change in cost?
38)
39) Referring to Scenario 3-7, what is the skewness statistic of the change
in the cost?
39)
40) Referring to Scenario 3-7, is the change in the cost right- or
left-skewed?
40)
41) Referring to Scenario 3-7, what is the kurtosis statistic of the change in
the cost?
41)
42) Referring to Scenario 3-7, is the change in the cost lepokurtic or
platykurtic?
42)
43) Referring to Scenario 3-7, what are the (absolute values of) the Z
scores of the change in cost?
43)
6
44) Referring to Scenario 3-7, are the data skewed? If so, how?
44)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers
the question.
45) In general, which of the following descriptive summary measures cannot be
easily approximated from a boxplot?
A) The variance
B) The median
C) The range
D) The interquartile range
45)
46) The smaller the spread of scores around the arithmetic mean,
A) the smaller the coefficient of variation.
B) the smaller the standard deviation.
C) the smaller the interquartile range.
D) All the above.
46)
47) In right-skewed distributions, which of the following is the correct
statement?
A) The distance from Q1 to Q2 is greater than the distance from Q2 to Q3.
B) The distance from Q1 to Q2 is less than the distance from Q2 to Q3.
C) The mode is greater than the arithmetic mean.
D) The arithmetic mean is less than the median.
47)
48) In left-skewed distributions, which of the following is the correct statement?
A) The distance from the smallest observation to Q2 is less than the
distance from Q2 to the largest observation.
B) The distance from Q1 to Q3 is twice the distance from the Q1 to Q2.
C) The distance from the smallest observation to Q1 is larger than the
48)
49) Which of the following is NOT sensitive to extreme values?
A) The standard deviation
B) The coefficient of variation
C) The range
D) The interquartile range
49)
distance from Q3 to the largest observation.
D) The distance from Q1 to Q2 is smaller than the distance from Q2 to Q3.
7
50) A manufacturer of ashlight ba eries took a sample of 130 ba eries from a
day's production and used them continuously until they were drained. The
number of hours until failure are recorded. Given below is the boxplot of the
number of hours it took to drain each of the 130 ba eries. The distribution of
the number of hours is
A) symmetrical.
C) left-skewed.
50)
B) right-skewed.
D) none of the above
SHORT ANSWER. Write the word or phrase that best completes each statement or answers
the question.
SCENARIO 3-1
Health care issues are receiving much attention in both academic and political arenas. A sociologist
recently conducted a survey of citizens over 60 years of age whose net worth is too high to qualify
for Medicaid. The ages of 25 senior citizens were as follows:
60 61 62 63 64 65 66 68 68 69 70 73 73
74 75 76 76 81 81 82 86 87 89 90 92
51) Referring to Scenario 3-1 determine the rst quartile of the ages of the
senior citizens.
51)
52) Referring to Scenario 3-1 determine the third quartile of the ages of
the senior citizens.
52)
8
53) Referring to Scenario 3-1, determine the interquartile range of the ages
of the senior citizens.
53)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers
the question.
54) Referring to Scenario 3-1, determine which of the following is the correct
statement.
A) The mean age of senior citizens sampled is 73.5 years of age.
B) The middle 50% of the senior citizens sampled are between 65.5 and
73.0 years of age.
C) One fourth of the senior citizens sampled are below 65.5 years of age.
D) All of the above are correct.
54)
55) Referring to Scenario 3-1, identify which of the following is the correct
statement.
A) The middle 50% of the senior citizens sampled are between 65.5 and
73.0 years of age.
B) 25% of the senior citizens sampled are older than 81.5 years of age.
C) One fourth of the senior citizens sampled are below 64 years of age.
D) All of the above are correct.
55)
56) True or False: In a set of numerical data, the value for Q3 can never be
56)
smaller than the value for Q1.
A) True
B) False
57) True or False: In a set of numerical data, the value for Q2 is always halfway
between Q1 and Q3.
A) True
B) False
57)
58) True or False: If the distribution of a data set were perfectly symmetrical, the
distance from Q1 to the median would always equal the distance from Q3 to
the median in a boxplot.
A) True
B) False
58)
59) True or False: In right-skewed distributions, the distance from Q3 to the
largest value is greater than the distance from the smallest observation to
Q1.
A) True
B) False
59)
9
60) True or False: In left-skewed distributions, the distance from the smallest
value to Q1 is greater than the distance from Q3 to the largest value.
A) True
B) False
60)
61) True or False: A boxplot is a graphical representation of a five-number
summary.
A) True
B) False
61)
62) True or False: The five-number summary consists of the smallest value, the
first quartile, the median, the third quartile, and the largest value.
A) True
B) False
62)
63) True or False: In a boxplot, the box portion represents the data between the
first and third quartile values.
A) True
B) False
63)
64) True or False: The line drawn within the box of the boxplot always
represents the arithmetic mean.
A) True
B) False
64)
65) True or False: The line drawn within the box of the boxplot always
represents the median.
A) True
B) False
65)
66) True or False: In exploratory data analysis, a boxplot can be used to illustrate
the median, quartiles, and extreme values.
A) True
B) False
66)
67) True or False: The interquartile range is a measure of variation or dispersion
in a set of data.
A) True
B) False
67)
68) True or False: The interquartile range is a measure of central tendency in a
set of data.
A) True
B) False
68)
10
SHORT ANSWER. Write the word or phrase that best completes each statement or answers
the question.
SCENARIO 3-3
The ordered array below represents the number of vitamin supplements sold by a health food
store in a sample of 16 days.
19, 19, 20, 20, 22, 23, 25, 26, 27, 30, 33, 34, 35, 36, 38, 41
Note: For this sample, the sum of the values is 448, and the sum of the squared di erences between
each value and the mean is 812.
69) Referring to Scenario 3-3, the rst quartile of the number of vitamin
supplements sold in this sample is ________.
69)
70) Referring to Scenario 3-3, the third quartile of the number of vitamin
supplements sold in this sample is ________.
70)
71) Referring to Scenario 3-3, the interquartile range of the number of
vitamin supplements sold in this sample is ________.
71)
72) Referring to Scenario 3-3, the ve-number summary of the data in
this sample consists of ________, ________, ________, ________,
________.
72)
73) Referring to Scenario 3-3, construct a boxplot for the data in this
sample.
73)
SCENARIO 3-5
The rates of return of a Fortune 500 company over the past 15 years are: 3.17%, 4.43%, 5.93%,
5.43%, 7.29%, 8.21%, 6.23%, 5.23%, 4.34%, 6.68%, 7.14%, -5.56%, -5.23%, -5.73%, -10.34%
74) Referring to Scenario 3-5, construct a boxplot for the rate of return.
11
74)
SCENARIO 3-7
In a recent academic year, many public universities in the United States raised tuition and fees due
to a decrease in state subsidies. The change in the cost of tuition, a shared dormitory room, and the
most popular meal plan from the previous academic year for a sample of 10 public universities
were as follows: $1,589, $593, $1,223, $869, $423, $1,720, $708, $1425, $922 and $308.
75) Referring to Scenario 3-7, what is the ve-number summary of the
change in the cost?
75)
76) Referring to Scenario 3-7, what is the interquartile range of the change
in the cost?
76)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers
the question.
77) In a perfectly symmetrical bell-shaped "normal" distribution
A) the arithmetic mean equals the mode.
B) the median equals the mode.
C) the arithmetic mean equals the median.
D) All the above.
77)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers
the question.
SCENARIO 3-9
The following table represents the assets in billions of dollars of the five largest bond funds
sometime in the past.
Bond Fund
Assets (Billions $)
PIMCO Total Return Fund
246
Vanguard Total Bond Market Index
Fund
110
Templeton Global Bond Fund
57
Vanguard Total Bond Market Index
Fund II
56
Vanguard Inflation-Protected
Securities
43
78) Referring to Scenario 3-9, what is the mean for this population of the
five largest bond funds?
12
78)
79) Referring to Scenario 3-9, what are the variance and standard
deviation for this population?
79)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers
the question.
80) According to the empirical rule, if the data form a "bell-shaped" normal
distribution, ________ percent of the observations will be contained within 2
standard deviations around the arithmetic mean.
A) 88.89
B) 68.26
C) 93.75
D) 95.44
80)
81) According to the empirical rule, if the data form a "bell-shaped" normal
distribution, ________ percent of the observations will be contained within 3
standard deviations around the arithmetic mean.
A) 95.0
B) 99.7
C) 68.26
D) 75.00
81)
82) According to the Chebyshev rule, at least 93.75% of all observations in any
data set are contained within a distance of how many standard deviations
around the mean?
A) 4
B) 3
C) 1
D) 2
82)
83) According to the Chebyshev rule, at least what percentage of the
observations in any data set are contained within a distance of 2 standard
deviations around the mean?
A) 75%
B) 67%
C) 88.89%
D) 95%
83)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers
the question.
SCENARIO 3-11
Given below are the closing prices for the Dow Jones Industrial Average (DJIA) and the Standard &
Poor's (S&P) 500 Index over a 10-week period sometime in the past.
Dow
Jones
S&P 500
10,42 10,11
10,47
10,59 11,21 10,93 11,13 10,31
1
0 9,862
5 9,920
2
3
3
4
6
1,379 1,356 1,343 1,410 1,389 1,463 1,529 1,499 1,516 1,355
84) Referring to Scenario 3-11, what is the sample covariance between
the DJIA and the S&P 500 index?
13
84)
85) Referring to Scenario 3-11, what is the sample correlation coe cient
between the DJIA and the S&P 500 index?
85)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers
the question.
86) Referring to Scenario 3-11, how will you classify the linear relationship
between the DJIA and the S&P 500 index?
A) Moderate
B) Weak
C) Strong
D) No relationship
86)
87) Referring to Scenario 3-11, for the week when the DJIA is high, you will
expect the S&P index in that week to
A) be low.
B) be about the same value as the DJIA.
C) be high.
D) have no relationship with the DJIA value.
87)
88) Referring to Scenario 3-11, you will expect an increase in the DJIA to be
associated with
A) no predictable change in the S&P 500 index.
B) an increase in the S&P 500 index.
C) a decrease in the S&P 500 index.
D) no predictable change in the DJIA.
88)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers
the question.
SCENARIO 3-13
Energy drink consumption has continued to gain in popularity since the 1997 debut of Red Bull, the
current leader in the energy drink market. Given below are the exam scores and the number of
12-ounce energy drinks consumed within a week prior to the exam of 10 college students.
Exam Scores
Number of
Drinks
75
92
84
64
64
86
81
61
73
93
5
3
2
4
2
7
3
0
1
0
89) Referring to Scenario 3-13, what is the sample covariance between the
exam scores and the number of energy drinks consumed?
14
89)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers
the question.
90) Referring to Scenario 3-13, how will you classify the linear relationship
between the exam scores and the number of energy drinks consumed?
A) Moderate
B) Strong
C) No relationship
D) Weak
90)
91) Referring to Scenario 3-13, for a student who has consumed a high number
of energy drinks within the week prior to the exam, you will expect his/her
exam score to
A) not be noticeably different than the exam score had he/she not
consumed as much.
B) be noticeably higher than the exam score had he/she not consume as
much energy drink.
C) not be noticeably different than the amount of energy drinks that
he/she consumed.
D) be noticeably lower than the exam score had he/she not consume as
much energy drink.
91)
92) Referring to Scenario 3-13, you will expect a decrease in the amount of
energy drink consumed within the week prior to the exam to be associated
with
A) an increase in the exam score.
B) no predictable change in the exam score.
C) no predictable change in the amount of energy drink consumed after
the exam.
D) a decrease in the exam score.
92)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers
the question.
93) Referring to Scenario 3-13, what is the sample correlation coe cient
between the exam scores and the number of energy drinks
consumed?
15
93)
Answer Key
Testname: CH3-NUMERICAL DESCRIPTIVE MEASUR ES
1) B
2) C
3) C
4) 74.04
years
5) 73 years
6) B
7) B
8) A
9) A
10) 28
11) 26.5
12) 2.48%
13) $978, $895.5
14) C
15) A
16) B
17) A
18) 0.36
19) Slightly positive or right-skewed
20) -1.01
21) lepokurtic
22) 94.96 years2
23) 9.74 years
24) 13.16%
25) B
26) 0.29
27) Right-skewed
28) -1.32
29) platykurtic
30) 22
31) 54.13
32) 7.36
33) 26.3%
34) Slightly positive or right-skewed
35) 18.55%
36) Left-skewed
37) $491.8
38) 50.29%
39) 0.22
40) Right-skewed
41) -1.32
42) platykurtic
43) 1.24, 0.78, 0.50, 0.22, 1.13, 1.51, 0.55, 0.91, 0.11, 1.36
44) skewed to the right
45) A
16
Answer Key
Testname: CH3-NUMERICAL DESCRIPTIVE MEASUR ES
46) D
47) B
48) C
49) D
50) B
51) 65.5
years
52) 81.5 years
53) 16 years
54) C
55) B
56) A
57) B
58) A
59) A
60) A
61) A
62) A
63) A
64) B
65) A
66) A
67) A
68) B
69) 20
70) 35
71) 15
72) 19, 20, 26.5, 35, 41
73)
17
Answer Key
Testname: CH3-NUMERICAL DESCRIPTIVE MEASUR ES
74)
75) $308,
76) $832
$593, $895.5, $1425, and $1720
77) D
78) $102.4
billions
79) 5,684.24 squared billion dollars, $75.39 billions, respectively
80) D
81) B
82) A
83) A
84) 31594.7333
85) 0.9274
86) C
87) C
88) B
89) 3.7667
90) D
91) A
92) B
93) 0.1450
18